1. PHY 752 Solid State Physics Reading: Chapter 4 in GGGPP
11-11:50 AM MWF Olin 107
Plan for Lecture 13:
Reading: Chapter 4 in GGGPP
Approximations to the many electron problem --Density functional theory
General theorem
Practical calculation scheme
Some results
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PHY 752 Fall Lecture 13

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PHY 752 Fall Lecture 13

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PHY 752 Fall Lecture 13

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PHY 752 Fall Lecture 13

5. Density functional theory
Describes the relationship between the many electron problem and independent electron treatments.
Proof of theorem
Estimates of F[n].
9/23/2015
PHY 752 Fall Lecture 13

6. Density functional theory -- continued
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PHY 752 Fall Lecture 13

7. Hohenberg and Kohn: formal proof of basic theorem
Kinetic energy
External potential
Coulomb interaction
9/23/2015
PHY 752 Fall Lecture 13

8. Theorem: The density n(r) of the ground state of the system is a unique functional of the external potential v(r).
Proof: Consider two Hamiltonians H and H’ differing only by external potentials v and v’.
9/23/2015
PHY 752 Fall Lecture 13

9. We can also show:
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PHY 752 Fall Lecture 13

10. The theorem implies that the ground state energy E can be considered as a functional of the density n(r)
Thus, the determination of the ground state energy E is transformed into a minimization of the functional with respect to the density n(r), transforming a many particle minimization into a single particle minimization.
In practice, the functional form of F[n] is not known, but if it were, we could use optimization methods to determine the ground state energy Ev[n].
9/23/2015
PHY 752 Fall Lecture 13

11. Kohn-Sham scheme to find ground state energy Ev[n]
Assume that the electron density can be expressed in terms of N independent electron orbitals
For a given external potential vext(r) the ground state energy is given by E
EH[n]=
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PHY 752 Fall Lecture 13

12. Kohn-Sham equations
where
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PHY 752 Fall Lecture 13

13. Estimate of the exchange-correlation contribution
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PHY 752 Fall Lecture 13

14. Digression on spatially varying electron densityj k l
2/13/2015
PHY 752 Spring Lecture 13

15. Correlation functionals Local density approximation (LDA)
9/23/2015
PHY 752 Fall Lecture 13

16. Interpolation function for LDA: Ec[n]=
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PHY 752 Fall Lecture 13

17. More complicated exchange-correlation functionals
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PHY 752 Fall Lecture 13

18. Some details of the Generalized Gradient Approximation
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PHY 752 Spring Lecture 13

19. Summary: Kohn-Sham formulation of density functional theory
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PHY 752 Spring Lecture 13

20. Self-consistent solution
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PHY 752 Spring Lecture 13

21. Kohn-Sham formulation of density functional theory
Results of self-consistent calculations
Some remaining issues
Theory for Eexc[n] still underdevelopment
This formalism does not access excited states
Strongly correlated electron systems are not well approximated
2/13/2015
PHY 752 Spring Lecture 13

22. Comparison of LDA and GGA binding energy curves
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PHY 752 Spring Lecture 13

23. LDA vs GGA performance wrt normal modes of vibration
A: B. N. Mavrin et al, J. Exp. Theor. Phys. 96,53 (2003); B: F. Harbach and F. Fischer, Phys. Status Solidi B 66, 237 (1974) – room temp. C: Ref. B at liquid nitrogen temp.; D: L. Popović et al, J. Raman Spectrosc. 34,77 (2003).
2/13/2015
PHY 752 Spring Lecture 13